Let $$g_{11} = \frac{1-x_2^2}{1-x_1^2-x_2^2},\ g_{22} = \frac{1-x_1^2}{1-x_1^2-x_2^2},\ g_{12} = \frac{x_1x_2}{1-x_1^2-x_2^2}$$ be a metric on the 2-dimension disk $D^2 = \{(x_1,x_2)|\ x_1^2+x_2^2 \le 1\}$. In fact, this metric is the pullback metric of the semisphere function: $$\phi : (x_1,x_2) \to (x_1,x_2,\sqrt{1-x_1^2-x_2^2})$$ where the metric on semisphere is induced by Euclid metric.
I'm trying to solve the following equation system: $$1 + (\frac{\partial h}{\partial x_1})^2 = g_{11}$$ $$1 + (\frac{\partial h}{\partial x_2})^2 = g_{22}$$ $$\frac{\partial h}{\partial x_1}\frac{\partial h}{\partial x_2} = g_{12}$$
For every solution $h$, the graph of $\{(x_1,x_2,h(x_1,x_2))\ |\ (x_1,x_2) \in D^2\}$ will be a isometric embedding of $(D^2,g)$ to $\mathbb{R}^3$, thus semisphere function is a solution. I'm trying to solve this equation system with some numerical method, and I got a solution looks like bending half of the semisphere, making its inside to outside, but keeping the other half unchanged.
My question is: Does semisphere admits such an isometric deformation? And are there some useful materials about isometric deformation of semisphere in $\mathbb{R}^3$?